Definition:Field (Abstract Algebra)
This page is about Field in the context of Abstract Algebra. For other uses, see Field.
Definition
Let $\struct {F, +, \times}$ be an algebraic structure.
Definition $1$
$\struct {F, +, \times}$ is a field if and only if:
- $(1): \quad$ the algebraic structure $\struct {F, +}$ is an abelian group
- $(2): \quad$ the algebraic structure $\struct {F^*, \times}$ is an abelian group where $F^* = F \setminus \set {0_F}$
- $(3): \quad$ the operation $\times$ distributes over $+$.
Definition $2$
Let $\struct {F, +, \times}$ be an integral domain such that every non-zero element $a$ of $F$ has a multiplicative inverse $a^{-1}$ such that:
- $a \times a^{-1} = 1_F = a^{-1} \times a$
where $1_F$ denotes the unity of $\struct {F, +, \times}$.
Then $\struct {F, +, \times}$ is a field.
Definition $3$
Let $\struct {F, +, \times}$ be a commutative ring with unity $\struct {F, +, \times}$ such that every non-zero element $a$ of $F$ has a multiplicative inverse:
- $a^{-1}$ such that $a \times a^{-1} = 1_F = a^{-1} \times a$
where $1_F$ denotes the unity of $\struct {F, +, \times}$.
Then $\struct {F, +, \times}$ is a field.
Definition $4$
Let $\struct {F, +, \times}$ be a non-trivial division ring whose ring product is commutative.
Then $\struct {F, +, \times}$ is a field.
Definition $5$
$\struct {F, +, \times}$ is a field if and only if it fulfils the conditions of the field axioms, as follows:
| \((\text A 0)\) | $:$ | Closure under addition | \(\ds \forall x, y \in F:\) | \(\ds x + y \in F \) | |||||
| \((\text A 1)\) | $:$ | Associativity of addition | \(\ds \forall x, y, z \in F:\) | \(\ds \paren {x + y} + z = x + \paren {y + z} \) | |||||
| \((\text A 2)\) | $:$ | Commutativity of addition | \(\ds \forall x, y \in F:\) | \(\ds x + y = y + x \) | |||||
| \((\text A 3)\) | $:$ | Identity element for addition | \(\ds \exists 0_F \in F: \forall x \in F:\) | \(\ds x + 0_F = x = 0_F + x \) | $0_F$ is called the zero | ||||
| \((\text A 4)\) | $:$ | Inverse elements for addition | \(\ds \forall x \in F: \exists x' \in F:\) | \(\ds x + x' = 0_F = x' + x \) | $x'$ is called a negative element | ||||
| \((\text M 0)\) | $:$ | Closure under product | \(\ds \forall x, y \in F:\) | \(\ds x \times y \in F \) | |||||
| \((\text M 1)\) | $:$ | Associativity of product | \(\ds \forall x, y, z \in F:\) | \(\ds \paren {x \times y} \times z = x \times \paren {y \times z} \) | |||||
| \((\text M 2)\) | $:$ | Commutativity of product | \(\ds \forall x, y \in F:\) | \(\ds x \times y = y \times x \) | |||||
| \((\text M 3)\) | $:$ | Identity element for product | \(\ds \exists 1_F \in F, 1_F \ne 0_F: \forall x \in F:\) | \(\ds x \times 1_F = x = 1_F \times x \) | $1_F$ is called the unity | ||||
| \((\text M 4)\) | $:$ | Inverse elements for product | \(\ds \forall x \in F^*: \exists x^{-1} \in F^*:\) | \(\ds x \times x^{-1} = 1_F = x^{-1} \times x \) | |||||
| \((\text D)\) | $:$ | Product is distributive over addition | \(\ds \forall x, y, z \in F:\) | \(\ds x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z} \) |
These are called the field axioms.
Addition
The distributand $+$ of a field $\struct {F, +, \times}$ is referred to as field addition, or just addition.
Product
The distributive operation $\times$ in $\struct {F, +, \times}$ is known as the (field) product.
Also defined as
Some sources do not insist that the field product of a field is commutative.
That is, what they define as a field, $\mathsf{Pr} \infty \mathsf{fWiki}$ defines as a division ring.
When they wish to refer to a field in which the field product is commutative, the term commutative field is used.
Examples
Field of rational numbers
The field of rational numbers $\struct {\Q, + \times, \le}$ is the set of rational numbers under the two operations of addition and multiplication, with an ordering $\le$ compatible with the ring structure of $\Q$.
Field of real numbers
The field of real numbers $\struct {\R, +, \times, \le}$ is the set of real numbers under the two operations of addition and multiplication, with an ordering $\le$ compatible with the ring structure of $\R$..
Field of complex numbers
The field of complex numbers $\struct {\C, +, \times}$ is the set of complex numbers under the two operations of addition and multiplication.
Field of integers modulo $p$
Let $p \in \Bbb P$ be a prime number.
Let $\Z_p$ be the set of integers modulo $p$.
Let $+_p$ and $\times_p$ denote addition modulo $p$ and multiplication modulo $p$ respectively.
The algebraic structure $\struct {\Z_p, +_p, \times_p}$ is the field of integers modulo $p$.
Smallest fields
The smallest field is the set of integers modulo $2$ under modulo addition and modulo multiplication:
- $\struct {\Z_2, +_2, \times_2}$
Also see
- Results about fields in the context of abstract algebra can be found here.
Linguistic Note
Note that while in English the word for this entity is field, its name in other European languages translates as body.
The original work was done by Richard Dedekind, who used the word Körper. When translating his work into English, Eliakim Moore mistakenly used the word field.
The translator into French did not make the same mistake.
Internationalization
Field is translated:
| In Catalan: | cos | (literally: body) | ||
| In Dutch: | lichaam | (literally: body) | ||
| In Dutch: | veld | (literally: field) | ||
| In French: | corps | (literally: body) | ||
| In German: | Körper | (literally: body) | ||
| In Russian: | Поле | (literally: field) | Pronounced: póh-lye | |
| In Spanish: | campo | (literally: field) | ||
| In Spanish: | cuerpo | (literally: body) |
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $0$ Zero
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $0$ Zero